Optimal design of a surface-mounted permanent magnet motor with multi-grade ferrite magnets to reduce torque pulsations

Abstract

This paper proposes an optimal design for a surface-mounted permanent magnet motor (SPMM) to reduce torque pulsations, including cogging torque and torque ripple, by using multi-grade ferrite magnets. Based on a conventional SPMM with single-grade ferrite magnets, the proposed SPMM is designed with four-grade ferrite magnets and then optimized to minimize torque pulsations by maintaining the required torque, utilizing the Kriging method and a genetic algorithm. The results obtained by the finite element analysis show that the optimized SPMM with multi-grade ferrite magnets exhibits improved airgap flux density distribution with highly reduced cogging torque and torque ripple by maintaining the same average torque, as compared to the conventional SPMM. Furthermore, the analysis of the working points for the multi-grade ferrite magnets reveals that the optimized SPMM has good durability against the irreversible demagnetization.

Keywords

Cogging torque ferrite magnets finite element analysis genetic algorithm optimal design surface-mounted permanent magnet motor torque ripple

1. Introduction

The surface-mounted permanent magnet motors (SPMMs) are being increasingly applied to various domestic and industrial applications, not only due to their high torque density and high efficiency, but also owing to their simple structure, simple manufacturing, and easy maintenance [1]. In general, for the low-power SPMMs, the ferrite magnets will be utilized to obtain a high torque density with a low cost. However, for high-performance applications, not only the high torque density but also the high torque quality is required. Hence, the torque pulsations, including cogging torque and torque ripple, are always the big concern when designing high-performance SPMMs.

At present, there are many papers focusing on the reduction of torque pulsations, a part of them concentrate on the motor control strategy [2], and the majority of them suggest motor topologies [3]. In particular, the cogging torque has been one of the main concerns of SPMM. Various methods have been reported for suppressing cogging torque, such as selecting proper stator slot and rotor pole combinations, the fractional number of slot per pole [4], skewing of the stator slots or rotor poles [5], teeth notching [6], and optimization of the slot or slot-opening shifting [7]. However, the low cogging torque does not absolutely lead to low torque pulsations, especially in the case where the effects of armature reaction are severe. Therefore, many methods aiming at designing a sinusoidal magnetic flux density distribution are proposed to reduce both no-load cogging torque and on-load torque ripple, such as magnet pole shape optimization [8], magnet pole pairs [9], and sinusoidal magnet poles [10]. However, these methods inevitably introduce manufacturing difficulty or performance degradation.

In this paper, an optimal design is proposed for the SPMM to reduce torque pulsations by using multi-grade ferrite magnets. The proposed SPMM is designed with four-grade standard ferrite magnets and optimized to obtain low cogging torque and torque ripple by maintaining the required torque using the Kriging method and a genetic algorithm. To verify the contribution, a conventional SPMM with single-grade ferrite magnets is adopted for performance comparison based on a finite element method (FEM). Finally, the working points of the multi-grade ferrite magnets in the optimized SPMM are analyzed to examine the durability against the magnet irreversible demagnetization.

Fig. 1.

Conventional SPMM (basic model). (a) Stator and windings. (b) Rotor with ferrite magnets.

Fig. 2.

The distribution of the B_r(θ) of the basic model along the circumference.

Table 1

Specifications of the investigated SPMMs

Items	Unit	Basic model	Proposed model
Stator slots/magnet poles	-	6/4
Stator outer diameter	mm	88
Air gap	mm	0.5
Rotor outer diameter	mm	50
Magnet thickness	mm	5
Rotor inner diameter	mm	15
Rotor stack length	mm	52

Fig. 3.

The rotor of the proposed model and configuration of ferrite magnets.

Fig. 4.

The distribution of the B_r(θ) of the proposed model along the circumference.

2. Motor modeling

2.1. Modeling of the conventional SPMM

The conventional SPMM with single-grade ferrite magnets (NMF_D9-0.43T), nominated as the basic model, is shown in Fig. 1. The cogging torque is caused by the tangential component of the interaction force between the PM on the rotor and the stator teeth when there is no current in the windings. The cogging torque of the basic model is given as $\begin{eqnarray}\displaystyle T_{cog}({\theta})={\displaystyle \frac{{\pi}N_{p}L}{4{\mu}_{0}}}(r_{2}^{2}-r_{1}^{2})\mathop{\sum }_{n=1}^{\infty }n{\lambda}_{nN_{p}}B_{rnN_{p}}\sin (nN_{p}{\theta}) & & \displaystyle\end{eqnarray}$ (1) where N_p is the least common multiple of the number of the rotor poles and the number of stator slots, L is the length of the armature core, r₁ and r₂ are the outer radius of the rotor and the inner radius of the stator, respectively, 𝜆_{nN
_p} is the coefficient related to the permeability of the air gap, B_{rnN
_p} is the air gap flux density when the motor is equivalent to no slots, θ is the angle in the circumferential direction. The distribution of the PM remanence density (Br (θ)) of the basic model along the circumference is shown in Fig. 2. The periodic function $B_{r}^{2}({\theta})$ which has a period of π∕p can be expressed as: $\begin{eqnarray} \displaystyle B_{r}^{2}({\theta})=\left\{ \begin{array}{@{} ll@{}}B_{r}^{2} & \displaystyle -{\displaystyle \frac{{\alpha}_{p}{\pi}}{2p}}\leq {\theta}\leq {\displaystyle \frac{{\alpha}_{p}{\pi}}{2p}}\\[13pt] 0 & \displaystyle -{\displaystyle \frac{{\pi}}{2p}}\leq {\theta}\lt -{\displaystyle \frac{{\alpha}_{p}{\pi}}{2p}},\quad {\displaystyle \frac{{\alpha}_{p}{\pi}}{2p}}\lt {\theta}\leq {\displaystyle \frac{{\pi}}{2p}} \end{array} \right. & & \displaystyle \end{eqnarray}$ (2) where B_r is the remanence of the PM, p is the number of the pole pairs, 𝛼_p is the pole arc coefficient of the motor. The Fourier expansion of $B_{r}^{2}({\theta})$ on [−π∕2p, π∕2p] can be expressed as: $\begin{eqnarray}\displaystyle B_{r}^{2}({\theta})={\alpha}_{p}B_{r}^{2}+\mathop{\sum }_{n_{1}=1}^{\infty }\frac{2}{{\pi}}B_{r}^{2}\sin (n_{1}{\alpha}_{p}{\pi})\cos (2n_{1}p{\theta}). & & \displaystyle\end{eqnarray}$ (3) When 2n₁p = nN_p, B_rnNp can be expressed as: $\begin{eqnarray}\displaystyle B_{rnN_{p}}={\displaystyle \frac{4p}{n{\pi}N_{p}}}B_{r}^{2}\sin \left(nN_{p}{\displaystyle \frac{{\alpha}_{p}{\pi}}{2p}}\right). & & \displaystyle\end{eqnarray}$ (4)

2.2. Modeling of the proposed SPMM

Fig. 5.

Optimal design process.

Fig. 6.

Design variables. (a) Variable definition (b) Convergence results.

The proposed SPMM models and the basic SPMM models use the same stator with concentrated windings. The rotor of proposed SPMM with four-grade ferrite magnets (1-NMF_12E-0.47T; 2-NMF_9D-0.45T; 3-NMF_6E-0.41T; 4-NMF_6G-0.389T) is shown in Fig. 3. The distribution of the PM remanence density (B_r(θ)) of the proposed SPMM along the circumference is shown in Fig. 4. The periodic function $B_{r}^{2}({\theta})$ which has a period of π∕p can be expressed as: $\begin{eqnarray} \displaystyle B_{r}^{2}({\theta})=\left\{ \begin{array}{@{} ll@{}}B_{r1}^{2} & \displaystyle -\frac{{\alpha}_{1}{\pi}}{2p}\leq {\theta}\leq {\displaystyle \frac{{\alpha}_{1}{\pi}}{2p}}\\ [13pt] B_{r2}^{2} & \displaystyle -{\displaystyle \frac{{\alpha}_{2}{\pi}}{2p}}\leq {\theta}\lt -{\displaystyle \frac{{\alpha}_{1}{\pi}}{2p}},{\displaystyle \frac{{\alpha}_{1}{\pi}}{2p}}\lt {\theta}\leq \frac{{\alpha}_{2}{\pi}}{2p}\\ [13pt] B_{r3}^{2} & \displaystyle -\frac{{\alpha}_{3}{\pi}}{2p}\leq {\theta}\lt -\frac{{\alpha}_{2}{\pi}}{2p},\frac{{\alpha}_{2}{\pi}}{2p}\lt {\theta}\leq \frac{{\alpha}_{3}{\pi}}{2p}\\ [13pt] B_{r4}^{2} & \displaystyle -\frac{{\alpha}_{p}{\pi}}{2p}\leq {\theta}\lt -{\displaystyle \frac{{\alpha}_{3}{\pi}}{2p}},\frac{{\alpha}_{3}{\pi}}{2p}\lt {\theta}\leq {\displaystyle \frac{{\alpha}_{p}{\pi}}{2p}}\\ [13pt] 0 & \displaystyle -{\displaystyle \frac{{\pi}}{2p}}\leq {\theta}\lt -\frac{{\alpha}_{p}{\pi}}{2p},{\displaystyle \frac{{\alpha}_{p}{\pi}}{2p}}\lt {\theta}\leq \frac{{\pi}}{2p} \end{array} \right. & & \displaystyle \end{eqnarray}$ (5) where B_r1 is the remanence of ferrite 1, B_r2 is the remanence of ferrite 2, B_r3 is the remanence of ferrite 3, B_r4 is the remanence of ferrite 4, 𝛼₁ is the pole arc coefficient corresponding to ferrite 1, 𝛼₂ is the pole arc coefficient corresponding to the sum of ferrite 1 and ferrite 2, 𝛼₃ is the pole arc coefficient corresponding to the sum of ferrite 1, ferrite 2 and ferrite 3. The Fourier expansion of $B_{r}^{2}({\theta})$ on [−π∕2p, π∕2p] can be expressed as: $\begin{eqnarray}\displaystyle B_{r}^{2}({\theta})=B_{r0}+\displaystyle \mathop{\sum }_{n_{1}=1}^{\infty }B_{rn_{1}}\cos (2n_{1}p{\theta}) & & \displaystyle\end{eqnarray}$ (6) where $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle B_{r0}={\alpha}_{1}B_{r1}^{2}+({\alpha}_{2}-{\alpha}_{1})B_{r2}^{2}+({\alpha}_{3}-{\alpha}_{2})B_{r3}^{2}+({\alpha}_{p}-{\alpha}_{3})B_{r4}^{2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle B_{rn_{1}}={\displaystyle \frac{2}{n_{1}{\pi}}}\big[B_{r4}^{2}\sin (n_{1}{\alpha}_{p}{\pi})+(B_{r3}^{2}-B_{r4}^{2})\sin (n_{1}{\alpha}_{3}{\pi})+(B_{r2}^{2}-B_{r3}^{2})\sin (n_{1}{\alpha}_{2}{\pi})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \qquad \qquad +∼(B_{r1}^{2}-B_{r2}^{2})\sin (n_{1}{\alpha}_{1}{\pi})\big].\nonumber\end{eqnarray}$

The magnets in both SPMM models keep the same volume size with radial magnetization, and the steel sheet NSSMC 50H470 is used for the ferromagnetic parts of all SPMMs. The specifications of the abovementioned SPMMs are listed in Table 1.

3. Optimal design

To obtain the superior performance, the optimal design by using the Kriging method and a genetic algorithm is conducted as shown in Fig. 5. First, the objective functions, constraints, and design variables are determined. Then the design of the experiment process is performed for the selection of sampling points, and the Kriging method is carried out for the approximation modeling. The genetic algorithm (GA) is to evaluate the fitness from the function developed by the Kriging model, thus to determine the optimal results by the population. Finally, the optimal design results are verified by FEM. The fitness function F (x) used in GA can be expressed as: $\begin{eqnarray}\displaystyle F(x)=f(x)+{\varepsilon}P(x) & & \displaystyle\end{eqnarray}$ (7) where f (x) is the multi-objective function, P (x) is the penalty function, ϵ is the penalty coefficient, and x is a vector of the design variables.

Fig. 7.

Comparison of airgap flux density.

Fig. 8.

The phase back EMFs of all models.

Fig. 9.

FFT analysis of back EMFs.

Fig. 10.

Comparison of cogging torques.

The objective functions for minimizing cogging torque and torque ripple are shown in (10). The cogging torque T_cogging is defined as: $\begin{eqnarray}\displaystyle T_{cogging}=T_{\max }-T_{\min } & & \displaystyle\end{eqnarray}$ (8) where T_max and T_min are the maximum cogging torque and minimum cogging torque under no-load conditions, respectively.

The torque ripple T_ripple is defined as follow: $\begin{eqnarray}\displaystyle T_{ripple}={\displaystyle \frac{T_{p-p}}{T_{ave}}}\times 100\% & & \displaystyle\end{eqnarray}$ (9) where T_p−p is the peak-to-peak value of the torque and T_ave is the average electromagnetic torque. The constraints are the fixed magnet volume and minimum torque as shown in (11). The design variables are determined as the angular width of the magnets (X₁, X₂, and X₃), as shown in Fig. 6(a) and (12).

Objective functions $\begin{eqnarray}\displaystyle \text{Minimize cogging torque and torque ripple}. & & \displaystyle\end{eqnarray}$ (10)

Constraints $\begin{eqnarray}\displaystyle \text{Magnet volume }=36∼\text{cm}^{3}∼\text{Torque}>0.596∼\text{Nm}. & & \displaystyle\end{eqnarray}$ (11)

Design variables $\begin{eqnarray}\displaystyle 6^{\circ }\leq X_{1}\leq 12^{\circ }\quad 8^{\circ }\leq X_{2}\leq 18^{\circ }\quad 10^{\circ }\leq X_{3}\leq 20^{\circ }. & & \displaystyle\end{eqnarray}$ (12)

The optimal design variables are obtained by the genetic algorithm as X₁ = 8.50°, X₂ = 15.28°, and X₃ = 17.98°, respectively. Figure 6(b) shows the convergence history plots for the design variables.

4. Performance comparison

Figure 7 compares the airgap flux density of the SPMMs. It shows that the sinusoidal property of proposed and optimized models are improved when compared to the basic model. The back electromotive forces (EMFs) in phase of each model are obtained when the motor operates at the rated speed with no load and compared in Fig. 8. The basic model exhibits a typical flat top wave, while the proposed and optimized models improve the waveform. The fast Fourier transform (FFT) analysis results of the back EMFs are shown in Fig. 9. The total harmonic distortion (THD) for the basic model, proposed model, and optimized model are 21.84%, 18.92%, and 19.20%, respectively, as summarized in Table 2.

Table 2
Performance analysis results by the FEM

Item Unit Basic model Proposed model Optimized model

Back EMF V 56.48 56.92 56.66

THD of back EMF % 21.84 18.92 19.20

Cogging torque Nm 0.097 0.075 0.072

Torque (Average) Nm 0.596 0.604 0.601

Torque ripple % 22.09 17.02 16.10

Item	Unit	Basic model	Proposed model	Optimized model
Back EMF	V	56.48	56.92	56.66
THD of back EMF	%	21.84	18.92	19.20
Cogging torque	Nm	0.097	0.075	0.072
Torque (Average)	Nm	0.596	0.604	0.601
Torque ripple	%	22.09	17.02	16.10

Fig. 11.

Comparison of the electromagnetic torques.

Fig. 12.

FFT analysis of the electromagnetic torque.

Fig. 13.

Operating range of working points.

Figure 10 shows the comparison of cogging torques, which indicates that the cogging torque of the proposed model and optimized model are greatly reduced by 22.68% and 25.77%, respectively when compared to that of the basic model. Figure 11 shows the comparison of electromagnetic torques, in which the torque is predicted by feeding the stator windings by sinusoidal current excitations with the current density of 4 Arms/mm². The torque ripple of the proposed model and optimized model are greatly reduced by 22.95% and 27.12%, respectively, when compared to that of the basic model. In particular, the average torque of the proposed model and optimized model are both greater than that of the basic model. And the torque values for all models are listed in Table 2. Furthermore, the FFT analysis of the electromagnetic torques is shown in Fig. 12. It reveals that the proposed model and the optimized model exhibit higher fundamental values, but lower harmonic components when compared to the basic model.

The analysis of working points for the multi-grade ferrite magnets of the optimized model at the rated current density 4 Arms/mm² is shown in Fig. 13. It reveals that all ferrite magnets in the optimized model work over the knee points as marked by the red line, which indicates that the optimized model has good durability against the magnet irreversible demagnetization.

5. Conclusion

This paper has proposed an optimal design of a SPMM to reduce cogging torque and torque ripple by using multi-grade ferrite magnets. The results obtained by the finite element analysis show that the optimized SPMM through the Kriging method and genetic algorithm has greatly reduced cogging torque by 25.77%, and torque ripple by 27.12% while maintaining the similar average torque as compared to the conventional SPMM. Furthermore, the analysis of the working points for the multi-grade ferrite magnets reveals that the optimized SPMM has good durability against the irreversible demagnetization. Therefore, the proposed SPMM using multi-grade ferrite magnets, without specific magnet shaping, has attractions for commercial applications where low torque pulsations are desired.

Footnotes

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China, under Grant 51707107 and 51737008, in part by the Young Scholars Program of Shandong University, China, under Grant 2018WLJH29, in part by the projects funded by the China Postdoctoral Science Foundation, under Grant 2017M612269 and 2018T110688, and in part by the Youth Talent Support Program of Chinese Society for Electrical Engineering, under Grant JLB-2019-113.

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Optimal design of a surface-mounted permanent magnet motor with multi-grade ferrite magnets to reduce torque pulsations

Abstract

Keywords

1. Introduction

2.1. Modeling of the conventional SPMM

Table 2 Performance analysis results by the FEM Item Unit Basic model Proposed model Optimized model Back EMF V 56.48 56.92 56.66 THD of back EMF % 21.84 18.92 19.20 Cogging torque Nm 0.097 0.075 0.072 Torque (Average) Nm 0.596 0.604 0.601 Torque ripple % 22.09 17.02 16.10

Footnotes

Acknowledgements

References

Table 2
Performance analysis results by the FEM

Item Unit Basic model Proposed model Optimized model

Back EMF V 56.48 56.92 56.66

THD of back EMF % 21.84 18.92 19.20

Cogging torque Nm 0.097 0.075 0.072

Torque (Average) Nm 0.596 0.604 0.601

Torque ripple % 22.09 17.02 16.10